1
00:00:02,088 --> 00:00:03,862
So we’ve seen that the logistic equation
2
00:00:03,862 --> 00:00:06,652
with r equals 4 shows the butterfly effect
3
00:00:06,652 --> 00:00:09,311
sensitive dependence on initial conditions
4
00:00:09,311 --> 00:00:12,476
two imperceptibly different starting points
5
00:00:12,476 --> 00:00:15,976
can lead a big differences in the orbit later on
6
00:00:15,976 --> 00:00:18,746
and we saw this by these of computers
7
00:00:18,746 --> 00:00:21,935
I used the computer to iterate a function
8
00:00:21,935 --> 00:00:23,903
or solve the differential equation
9
00:00:23,903 --> 00:00:26,769
and we saw these diverging trajectories
10
00:00:26,769 --> 00:00:30,150
so I argued that this makes these systems
11
00:00:30,150 --> 00:00:34,337
even though they’re deterministic essentially unpredictable
12
00:00:34,337 --> 00:00:37,252
because we need a possible accuracy in order to do long term
13
00:00:37,252 --> 00:00:40,245
where often even medium term prediction
14
00:00:40,245 --> 00:00:45,986
in addition to posing a challenge to our predictions of real processes
15
00:00:45,986 --> 00:00:48,163
it also posses a challenge or puzzle
16
00:00:48,163 --> 00:00:51,733
for how we think about the computer results themselves
17
00:00:51,733 --> 00:00:55,937
since computers don’t store numbers to infinite precision
18
00:00:55,937 --> 00:00:58,962
and that’s what we need in order to do
19
00:00:58,962 --> 00:01:01,902
an accurate long term prediction on the computer
20
00:01:01,902 --> 00:01:05,105
we need enormous precision in the starting point
21
00:01:05,105 --> 00:01:07,559
and in all the numbers along the way
22
00:01:07,559 --> 00:01:10,302
so given that computers make tiny round off errors
23
00:01:10,302 --> 00:01:14,471
due to how they represents decimals
24
00:01:14,471 --> 00:01:19,165
we might wonder can we trust the result of computer simulations at all.
25
00:01:19,165 --> 00:01:20,850
and the answer turns out to be yes
26
00:01:20,850 --> 00:01:23,558
and there is a nice result that illustrates this
27
00:01:23,558 --> 00:01:25,448
known as the shadowing lemma.
28
00:01:25,448 --> 00:01:27,233
so what I want to do in this video is present
29
00:01:27,233 --> 00:01:29,713
the basic idea behind the shadowing lemma
30
00:01:29,713 --> 00:01:32,730
partly so you feel little better about the computer results
31
00:01:32,730 --> 00:01:35,216
but also because it’s just a neat result and a fun way
32
00:01:35,216 --> 00:01:37,376
and another way to think about
33
00:01:37,376 --> 00:01:39,285
what’s sensitive dependence on initial conditions
34
00:01:39,285 --> 00:01:42,495
and chaotic dynamics mean.
35
00:01:45,262 --> 00:01:47,938
so here is the idea behind shadowing
36
00:01:47,938 --> 00:01:50,918
Let’s say we’re studying an iterated function
37
00:01:50,918 --> 00:01:52,916
like the logistic equation
38
00:01:52,916 --> 00:01:56,646
and we choose an initial condition and then we compute
39
00:01:56,646 --> 00:01:58,901
we use a computer to compute an orbit
40
00:01:58,901 --> 00:02:00,714
and we could plot it in a time series plot
41
00:02:00,714 --> 00:02:02,900
just like I was having the computer do
42
00:02:02,900 --> 00:02:06,578
and maybe it looks something like this
43
00:02:06,578 --> 00:02:08,533
so the problem is that
44
00:02:08,533 --> 00:02:13,853
the computer is making round off errors
45
00:02:13,853 --> 00:02:15,918
it doesn’t have infinite precision
46
00:02:15,918 --> 00:02:19,100
and so because of sensitive dependence on initial conditions
47
00:02:19,100 --> 00:02:23,280
this computed orbit is not actually the true orbit
48
00:02:23,280 --> 00:02:27,660
for the initial condition that we chose
49
00:02:27,660 --> 00:02:31,419
so initially I’ll drive in blue
50
00:02:31,419 --> 00:02:34,669
the true orbit with this initial condition
51
00:02:34,669 --> 00:02:38,953
I start here it would be close
52
00:02:44,203 --> 00:02:50,738
but then this is sensitive dependence on initial conditions
53
00:02:50,738 --> 00:02:56,208
the true orbit might depart from the computed orbit
54
00:03:07,065 --> 00:03:09,128
so I wanted the computer to tell me
55
00:03:09,128 --> 00:03:11,928
the orbit with this initial condition
56
00:03:11,928 --> 00:03:13,306
and what the computer told me
57
00:03:13,306 --> 00:03:16,926
is showing here in this black curve
58
00:03:16,926 --> 00:03:20,469
however this black curve the computed orbit is not
59
00:03:20,469 --> 00:03:22,935
the true orbit for this initial condition
60
00:03:22,935 --> 00:03:27,564
the computer makes small round off errors
61
00:03:27,564 --> 00:03:31,434
in how it is storing decimal numbers
62
00:03:31,434 --> 00:03:35,003
and so the computed orbit is not the same as a true orbit
63
00:03:35,003 --> 00:03:38,253
this is another manifestation on the butterfly effect
64
00:03:38,253 --> 00:03:41,194
so at this point we can ask if this black curve,
65
00:03:41,194 --> 00:03:44,679
this black time series is not the true orbit, what is it?
66
00:03:44,679 --> 00:03:46,137
does it have any meaning at all
67
00:03:46,137 --> 00:03:50,084
or is it just garbage just some sort of randomness
68
00:03:50,084 --> 00:03:52,550
and amazingly it turns out that
69
00:03:52,550 --> 00:03:56,222
this computed orbit does have some meaning
70
00:03:56,222 --> 00:04:00,295
it’s the true orbit for some other initial condition
71
00:04:00,295 --> 00:04:05,721
so let me draw, picture of what that might look like
72
00:04:05,721 --> 00:04:12,028
the risk of clouding this up
73
00:04:24,057 --> 00:04:29,618
ok, so the black time series is what we computed.
74
00:04:29,618 --> 00:04:33,601
the blue is the true orbit for this initial condition
75
00:04:33,601 --> 00:04:37,825
the red is some other true orbit
76
00:04:49,579 --> 00:04:53,410
so this black curve is indeed an orbit of the logistic equation
77
00:04:53,410 --> 00:04:55,395
or whatever it is we’re studying
78
00:04:55,395 --> 00:04:57,650
it just happens to not be the exact orbit
79
00:04:57,650 --> 00:05:00,466
or the true orbit for initial condition that I thought
80
00:05:00,466 --> 00:05:04,673
but it’s a true orbit for some other initial condition
81
00:05:04,673 --> 00:05:07,812
so I can still interpret this black time series
82
00:05:07,812 --> 00:05:10,232
as a trajectory of the logistic equation
83
00:05:10,232 --> 00:05:12,206
it’s not garbage, it’s not nonsense.
84
00:05:12,206 --> 00:05:14,715
it just happens to not be the exact trajectory
85
00:05:14,715 --> 00:05:18,250
I maybe thought I was initially studying
86
00:05:18,250 --> 00:05:20,917
so this says that numerical results
87
00:05:20,917 --> 00:05:23,892
or at least iterated maps like the logistic equation
88
00:05:23,892 --> 00:05:27,501
even though round off error precision
89
00:05:27,501 --> 00:05:29,999
in the computer finite precision in the computer
90
00:05:29,999 --> 00:05:34,758
and the butterfly effect means that this black computed orbit
91
00:05:34,758 --> 00:05:39,572
it’s not true a faithful representation
92
00:05:39,572 --> 00:05:41,666
of the orbit for this initial condition
93
00:05:41,666 --> 00:05:46,789
it’s still faithful representation of an orbit of the logistic equation
94
00:05:50,095 --> 00:05:54,081
so we would say that this computed orbit in black
95
00:05:54,081 --> 00:05:56,481
shadows this red orbit
96
00:05:56,481 --> 00:05:59,736
it might not be exactly this true orbit
97
00:05:59,736 --> 00:06:04,786
but it’s arbitrarily close to some true orbit
98
00:06:04,786 --> 00:06:10,505
and this result that this is true is know as the shadowing lemma
99
00:06:10,505 --> 00:06:14,986
and a lemma in mathematics is a result that’s used as an intermediate step
100
00:06:14,986 --> 00:06:19,028
to prove or demonstrate some other central or more important result
101
00:06:19,028 --> 00:06:21,888
in any event this is a pretty famous result
102
00:06:21,888 --> 00:06:24,320
and it’s known as the shadowing lemma
103
00:06:24,320 --> 00:06:26,035
and the phenomenon is shadowing
104
00:06:26,035 --> 00:06:30,877
shadowing is when a computed orbit which is in a sense wrong
105
00:06:30,877 --> 00:06:33,211
due to the butterfly effect and finite precision
106
00:06:33,211 --> 00:06:38,932
nevertheless shadows comes arbitrarily close to some other true orbit
107
00:06:41,632 --> 00:06:44,352
so shadowing is I think it’s a strange phenomena
108
00:06:44,352 --> 00:06:47,167
and it’s fun and interesting to think about
109
00:06:47,167 --> 00:06:53,263
Let me give an analogy to illustrate this idea behind shadowing
110
00:06:53,263 --> 00:06:58,902
so let’s say you asked me to draw a portrait of somebody
111
00:06:58,902 --> 00:07:01,377
and as you know because you’re seeing my draw
112
00:07:01,377 --> 00:07:03,890
I’m actually not very good at drawing at all
113
00:07:03,890 --> 00:07:06,522
so I try to draw a true accurate portrait of
114
00:07:06,522 --> 00:07:11,829
whomever you asked me to make a portrait of
115
00:07:11,829 --> 00:07:13,505
but since I’m not very good
116
00:07:13,505 --> 00:07:15,751
I make a little mistake around the eyes
117
00:07:15,751 --> 00:07:17,475
and I make a little mistake around the mouth
118
00:07:17,475 --> 00:07:19,451
and I make a little mistake around the nose
119
00:07:19,451 --> 00:07:21,902
I make little mistake actually kind of all the time
120
00:07:21,902 --> 00:07:25,183
because I’m just not very very good at drawing
121
00:07:25,183 --> 00:07:30,732
so the result is I hand you a portrait of that I’ve drawn
122
00:07:30,732 --> 00:07:32,241
and you would look at it
123
00:07:32,241 --> 00:07:35,255
and you’d say that looks nothing like the person
124
00:07:35,255 --> 00:07:37,335
you were supposed to be drawing
125
00:07:37,335 --> 00:07:40,350
and I would say, yeah you’re right sorry
126
00:07:40,350 --> 00:07:45,033
but in the shadowing lemma picture I’ve drawn
127
00:07:45,033 --> 00:07:47,989
yes what I’ve drawn is not an accurate picture of the person
128
00:07:47,989 --> 00:07:50,991
I was trying to draw or you wanted me to draw
129
00:07:50,991 --> 00:07:53,460
but nevertheless what I’ve drawn is
130
00:07:53,460 --> 00:07:56,339
an accurate portrait of somebody else
131
00:07:56,339 --> 00:07:59,357
so I haven’t drawn an accurate picture
132
00:07:59,357 --> 00:08:01,315
of your friend you wanted portrait of
133
00:08:01,315 --> 00:08:03,820
but of the six or seven billion people in the world
134
00:08:03,820 --> 00:08:09,129
I got this just very very close to right for somebody else
135
00:08:09,129 --> 00:08:11,388
so that’s the idea behind the shadowing lemma
136
00:08:11,388 --> 00:08:13,934
you ask the computer to not draw portrait
137
00:08:13,934 --> 00:08:17,312
but make time series of the particular initial condition
138
00:08:17,312 --> 00:08:20,178
and the computer makes little errors
139
00:08:20,178 --> 00:08:22,203
because it just has finite precision
140
00:08:22,203 --> 00:08:23,685
it’s certainly better at arithmetic than
141
00:08:23,685 --> 00:08:26,261
I am at drawing but it still has finite precision
142
00:08:26,261 --> 00:08:30,071
and so it hands you back not a portrait but a time series
143
00:08:30,071 --> 00:08:33,015
and this time series is not exactly what you wanted
144
00:08:33,015 --> 00:08:36,738
it’s not the true exact time series for the initial condition
145
00:08:36,738 --> 00:08:42,264
but it’s a it is true or arbitrarily close to true
146
00:08:42,264 --> 00:08:47,755
an arbitrarily close to true time series for some other initial condition
147
00:08:47,755 --> 00:08:50,968
so maybe you will be a little bit disappointed
148
00:08:50,968 --> 00:08:53,693
but nevertheless what the computer has given you
149
00:08:53,693 --> 00:08:57,405
does say something true about the dynamical system you’re studying
150
00:08:57,405 --> 00:09:03,327
it is arbitrarily close to a true orbit of the dynamical system
151
00:09:07,383 --> 00:09:10,427
I can’t prove the shadowing lemma in this course
152
00:09:10,427 --> 00:09:11,921
it’s a pretty technical result
153
00:09:11,921 --> 00:09:14,985
and we just don’t have the mathematical machinery to do it
154
00:09:14,985 --> 00:09:17,370
but let me say a few things that
155
00:09:17,370 --> 00:09:20,996
maybe will make it seem at least a little bit more plausible
156
00:09:20,996 --> 00:09:24,121
Ok, so let’s imagine that the dynamical system
157
00:09:24,121 --> 00:09:27,927
we’re interested in is not a deterministic iterated function
158
00:09:27,927 --> 00:09:31,748
but is a fair coin something that's random
159
00:09:31,748 --> 00:09:34,742
that has an element of chance in it.
160
00:09:34,742 --> 00:09:37,000
so that just every time you toss the coin
161
00:09:37,000 --> 00:09:41,140
with an equal probability it comes up heads or tails
162
00:09:41,140 --> 00:09:42,984
and then you asked me to say
163
00:09:42,984 --> 00:09:46,083
ok what are the next five outcomes going to be
164
00:09:46,083 --> 00:09:49,499
so that’s in a sense the orbit for this system
165
00:09:49,499 --> 00:09:51,726
and I might say oh I think it’s going to be
166
00:09:51,726 --> 00:09:55,115
heads heads tails tails tails
167
00:09:55,115 --> 00:09:58,805
and then the system ran and it turns out I was wrong
168
00:09:58,805 --> 00:10:01,430
not surprisingly it’s hard to predict
169
00:10:01,430 --> 00:10:03,795
perfectly random fair coin processes
170
00:10:03,795 --> 00:10:06,580
so anyway so I said heads heads tails tails tails
171
00:10:06,580 --> 00:10:08,020
and I’m wrong
172
00:10:08,020 --> 00:10:10,888
but then I could say yeah alright I’m wrong
173
00:10:10,888 --> 00:10:12,671
but I’m not totally wrong
174
00:10:12,671 --> 00:10:14,034
because if you wait long enough
175
00:10:14,034 --> 00:10:16,651
you’re going to see heads heads tails tails tails for sure
176
00:10:16,651 --> 00:10:20,341
because all possible outcomes of heads and tails are equally likely
177
00:10:20,341 --> 00:10:24,563
so yeah I told you a wrong result in this particular instance
178
00:10:24,563 --> 00:10:28,479
but it’s true in that you really will see heads heads tails tails tails
179
00:10:28,479 --> 00:10:31,199
so I’m not as wrong as you think I am
180
00:10:31,199 --> 00:10:33,679
ok so that’s not surprising
181
00:10:33,679 --> 00:10:39,790
or very deep statement about a random process like tossing coins
182
00:10:39,790 --> 00:10:43,354
what’s interesting is the same sort of argument holds
183
00:10:43,354 --> 00:10:48,942
for this deterministic system an iterated function
184
00:10:48,942 --> 00:10:52,754
so you asked me to predict the next five or the next fifth orbits
185
00:10:52,754 --> 00:10:56,963
and I and my calculator on the computer give you an answer
186
00:10:56,963 --> 00:11:00,155
and it turns out that that answer is wrong compared to the true orbit
187
00:11:00,155 --> 00:11:03,084
but I could say yeah ok sorry I got it wrong
188
00:11:03,084 --> 00:11:05,923
but I guarantee you that this if you wait long enough
189
00:11:05,923 --> 00:11:07,800
you would see this orbit so I’ve given you
190
00:11:07,800 --> 00:11:10,541
something that’s actually true
191
00:11:10,541 --> 00:11:15,891
so in a sense if a system is random or mixed up enough
192
00:11:15,891 --> 00:11:22,509
then errors made in trying to predict the system
193
00:11:22,509 --> 00:11:27,199
in this context can still produce results
194
00:11:27,199 --> 00:11:30,398
that could have been produced by the system
195
00:11:30,398 --> 00:11:32,499
and maybe another way to think about this is
196
00:11:32,499 --> 00:11:35,062
the things I’m calling errors are not errors
197
00:11:35,062 --> 00:11:38,641
in the sense that I forget the rules of arithmetic
198
00:11:38,641 --> 00:11:41,961
or think that you know I think that two plus two is five
199
00:11:41,961 --> 00:11:45,837
they’re just you know they are very small imprecisions
200
00:11:45,837 --> 00:11:49,769
and then it’s the butterfly effect that amplifies those imprecisions
201
00:11:49,769 --> 00:11:53,643
so in effect that we have these tiny errors
202
00:11:53,643 --> 00:11:57,471
but then the macroscopic manifestation of these errors
203
00:11:57,471 --> 00:12:00,343
is result of the dynamics of the system itself
204
00:12:00,343 --> 00:12:03,851
so in a sense that the errors or imprecisions
205
00:12:03,851 --> 00:12:05,562
I’m not even sure what word to use
206
00:12:05,562 --> 00:12:08,013
arise from the dynamics of the system
207
00:12:08,013 --> 00:12:11,789
not from some horrible blender or external source
208
00:12:11,789 --> 00:12:15,052
and so maybe it’s not surprising that the system induced errors
209
00:12:15,052 --> 00:12:20,409
or imprecisions are nevertheless in some way true to the system
210
00:12:20,409 --> 00:12:24,818
ok, so again this is certainly not a proof of the phenomena of shadowing
211
00:12:24,818 --> 00:12:28,669
but maybe these remarks help to make
212
00:12:28,669 --> 00:12:32,078
shadowing seem a little bit more plausible
213
00:12:32,078 --> 00:12:36,590
and as maybe you could tell in the last couple minutes
214
00:12:36,590 --> 00:12:38,228
there are different notions of randomness
215
00:12:38,228 --> 00:12:40,896
that might be appearing here is a process random
216
00:12:40,896 --> 00:12:44,358
if it’s made by a random process can a deterministic process
217
00:12:44,358 --> 00:12:45,782
have a random outcome
218
00:12:45,782 --> 00:12:48,293
so there’s some ideas that we want to unpack here
219
00:12:48,293 --> 00:12:51,050
and think more carefully about what one means
220
00:12:51,050 --> 00:12:53,332
when one says something is random
221
00:12:53,332 --> 00:12:56,054
so this will be the topic what is randomness
222
00:12:56,054 --> 00:12:57,487
and how do we think about it
223
00:12:57,487 --> 00:13:00,848
in dynamical systems of the next set of lectures.